The study of this geometry is parametrised by rotation number and critical exponents. In [1]}, for pairs \((\ell , \ell )\)  ; \( \ell >1\)  and \(\rho \)  irrational number of bounded type (i.e. \(\max _na_n<\infty \) ), the authors found a transition between degenerate geometry and bounded
geometry. In fact, they show under (REF ) that, if \( 1<\ell \le 2\)  and \(\rho \in {\mathbb {R}}\setminus {\mathbb {Q}}\) , then the geometry is degenerate and it is bounded (independently of (REF )) if \( \ell >2\)  and \(\rho \)  is irrational number of bounded type. In [2]}, for \( \ell >1\) , the author proved that the class of function \(f\)  of critical exponents \((1, \ell )\)  or \((\ell , 1)\)  have a degenerate geometry. In [3]}, the authors show that, for the maps in \({L} \)  with Fibonacci rotation number, when the critical exponents \((\ell _1,\ell _2)\)  belong in \((1,2)^2\) , the geometry is degenerate. Let us note that,
Differently from other previous works, information on the geometry of the system is obtained by the study of the asymptotic behaviour of the renormalization operator.
